# LMIs in Control/pages/Delay Dependent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

**The System**Edit

For this particular problem, suppose that we were given the time-delayed system in the form of:

where

Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

**The Data**Edit

In order to obtain the LMI, we need the following 3 matrices: .

**The Optimization Problem**Edit

Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where such that the closed-loop system:

is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

**The LMI:** The Delay-Dependent Stabilization of Time-Delay SystemsEdit

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar , a symmetric matrix and a matrix that satisfy the following:

where

**Conclusion:**Edit

Given the resulting stabilizing control gain matrix , it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

**Implementation**Edit

- Example Code - A GitHub link that contains code (titled "DelayDependentTimeDelay.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

**Related LMIs**Edit

- Delay Independent Time-Delay Stabilization - Equivalent LMI for delay-independent time-delay stabilization.

## External LinksEdit

A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.